interpolation

interpolation

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  MATHEMATICAL METHODS INTERPOLATION I YEAR B.Tech By Mr. Y. Prabhaker Reddy Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad.
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  SYLLABUS OF MATHEMATICALMETHODS (as per JNTU Hyderabad) Name of the Unit Name of the Topic Matrices and Linear system of equations: Elementary row transformations – Rank Unit-I – Echelon form, Normal form – Solution of Linear Systems – Direct Methods – LU Solution of Linear Decomposition from Gauss Elimination – Solution of Tridiagonal systems – Solution systems of Linear Systems. Eigen values, Eigen vectors – properties – Condition number of Matrix, Cayley – Unit-II Hamilton Theorem (without proof) – Inverse and powers of a matrix by Cayley – Eigen values and Hamilton theorem – Diagonalization of matrix Calculation of powers of matrix – – Eigen vectors Model and spectral matrices. Real Matrices, Symmetric, skew symmetric, Orthogonal, Linear Transformation - Orthogonal Transformation. Complex Matrices, Hermition and skew Hermition Unit-III matrices, Unitary Matrices - Eigen values and Eigen vectors of complex matrices and Linear their properties. Quadratic forms - Reduction of quadratic form to canonical form, Transformations Rank, Positive, negative and semi definite, Index, signature, Sylvester law, Singular value decomposition. Solution of Algebraic and Transcendental Equations- Introduction: The Bisection Method – The Method of False Position – The Iteration Method - Newton –Raphson Method Interpolation:Introduction-Errors in Polynomial Interpolation - Finite Unit-IV differences- Forward difference, Backward differences, Central differences, Symbolic Solution of Non- relations and separation of symbols-Difference equations – Differences of a linear Systems polynomial - Newton’s Formulae for interpolation - Central difference interpolation formulae - Gauss Central Difference Formulae - Lagrange’s Interpolation formulae- B. Spline interpolation, Cubic spline. Unit-V Curve Fitting: Fitting a straight line - Second degree curve - Exponential curve - Curve fitting & Power curve by method of least squares. Numerical Numerical Integration: Numerical Differentiation-Simpson’s 3/8 Rule, Gaussian Integration Integration, Evaluation of Principal value integrals, Generalized Quadrature. Unit-VI Solution by Taylor’s series - Picard’s Method of successive approximation- Eule Numerical Method -Runge kutta Methods, Predictor Corrector Methods, Adams- Bashforth solution of ODE Method. Determination of Fourier coefficients - Fourier series-even and odd functions - Unit-VII Fourier series in an arbitrary interval - Even and odd periodic continuation - Half- Fourier Series range Fourier sine and cosine expansions. Unit-VIII Introduction and formation of PDE by elimination of arbitrary constants and Partial arbitrary functions - Solutions of first order linear equation - Non linear equations - Differential Method of separation of variables for second order equations - Two dimensional Equations wave equation.
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  CONTENTS UNIT-IV(b) INTERPOLATION  Introduction  Introduction to Forward, Back ward and Central differences  Symbolic relations and Separation of Symbols  Properties  Newton’s Forward Difference Interpolation Formulae  Newton’s Backward Difference Interpolation Formulae  Gauss Forward Central Difference Interpolation Formulae  Gauss Backward Central Difference Interpolation Formulae  Striling’s Formulae  Lagrange’s Interpolation
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  INTERPOLATION The process offinding the curve passing through the points is called as Interpolation and the curve obtained is called as Interpolating curve. Interpolating polynomial passing through the given set of points is unique. Let be given set of observations and be the given function, then the method to find is called as an Interpolation. If is not in the range of and , then the method to find is called as Extrapolation. Equally Spaced Unequally Spaced Arguments Arguments Newton’s & Gauss Lagranges Interpolation Interpolation The Interpolation depends upon finite difference concept. If be given set of observations and let be their corresponding values for the curve , then is called as finite difference. When the arguments are equally spaced i.e. then we can use one of the following differences. Forward differences Backward differences Central differences Forward Difference Let us consider be given set of observations and let are corresponding values of the curve , then the Forward difference operator is denoted by and is defined as . In this case are called as First Forward differences of . The difference of first forward differences will give us Second forward differences and it is denoted by and is defined as
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  Similarly, the differenceof second forward differences will give us third forward difference and it is denoted by . Forward difference table First Forward Second Forward Third Forward Fourth differences differences differences differences . . . . . . . . . Note: If is common difference in the values of and be the given function then . Backward Difference Let us consider be given set of observations and let are corresponding values of the curve , then the Backward difference operator is denoted by and is defined as . In this case are called as First Backward differences of . The difference of first Backward differences will give us Second Backward differences and it is denoted by and is defined as Similarly, the difference of second backward differences will give us third backward difference and it is denoted by .
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  Backward difference table First Backward Second Backward Third Backward Fourth differences differences differences differences . . . . . . . . . . Note: If is common difference in the values of and be the given function then . Central differences Let us consider be given set of observations and let are corresponding values of the curve , then the Central difference operator is denoted by and is defined as  If is odd :  If is even : and The Central difference table is shown below Note: Let be common difference in the values of and be given function then
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  Symbolic Relations andSeparation of Symbols Average Operator: The average operator is defined by the equation (Or) Let is the common difference in the values of and be the given function, then the average operator is denoted by and is defined as Shift Operator: The Shift operator is defined by the equation Similarly, (Or) Let is the common difference in the values of and be the given function, then the shift operator is denoted by and is defined as Inverse Operator: The Inverse Operator is defined as In general, Properties 1) Prove that 2) Prove that Sol: Consider R.H.S: Sol: Consider L.H.S: 3) Prove that Sol: Case (i) Consider Case (ii) Consider Hence from these cases, we can conclude that 4) Prove that Sol: Consider
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  Hence 5) Prove that (Hint: Consider ) 6) Prove that 7) Prove that 8) Prove that Sol: We know that Sol: We know that Hence the result Hence proved that 9) Prove that Sol: We know that Squaring on both sides, we get L.H.S Hence the result Relation between the operator and Here Operator We know that Expanding using Taylor’s series , we get
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  Newton’s Forward InterpolationFormula Statement: If are given set of observations with common difference and let are their corresponding values, where be the given function then where Proof: Let us assume an degree polynomial ---> (i) Substitute in (i), we get Substitute in (i), we get Substitute in (i), we get Similarly, we get Substituting these values in (i), we get ----(ii) But given Similarly, , Substituting in the Equation (ii), we get
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  Newton’s Backward InterpolationFormula Statement: If are given set of observations with common difference and let are their corresponding values, where be the given function then where Proof: Let us assume an degree polynomial --> (i) Substitute in (i), we get Substitute in (i), we get Substitute in (i), we get Similarly, we get Substituting these values in (i), we get ---- (ii) But given Similarly, , Substituting in the Equation (ii), we get
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  Gauss forward centraldifference formula Statement: If are given set of observations with common difference and let are their corresponding values, where be the given function then where . Proof: Let us assume a polynomial equation by using the arrow marks shown in the above table. Let ---- ( 1 ) where are unknowns --- ( 2 ) Now, Therefore, ----- ( 3 ) and ----- ( 4 ) Substituting 2, 3, 4 in 1, we get Comparing corresponding coefficients, we get
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  , , Similarly, Substituting all these values of in (1), we get Gauss backward central difference formula Statement: If are given set of observations with common difference and let are their corresponding values, where be the given function then where . Proof: Let us assume a polynomial equation by using the arrow marks shown in the above table. Let ---- ( 1 ) where are unknowns --- ( 2 ) Now,
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  Therefore, ----- ( 3 ) ----- ( 4 ) Also Now, ----- ( 5 ) Substituting 2, 3, 4, 5 in 1, we get Comparing corresponding coefficients, we get , Also, Similarly, ,... Substituting all these values of in (1), we get , Stirling’s Formulae Statement: If are given set of observations with common difference and let are their corresponding values, where be the given function then where Proof: Stirling’s Formula will be obtained by taking the average of Gauss fo rward difference formula and Gauss Backward difference formula. We know that, from Gauss forward difference formula ---- > (1) Also, from Gauss backward difference formula ---- > (2) Now,
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  Lagrange’s Interpolation Formula Statement:If are given set of observations which are need not be equally spaced and let are their corresponding values, where be the given function then Proof: Let us assume an degree polynomial of the form ---- (1) Substitute , we get Again, , we get Proceeding like this, finally we get, Substituting these values in the Equation (1), we get Note: This Lagrange’s formula is used for both equally spaced and unequally spaced arguments.